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The US Particle Accelerator School Boston University June 16th, 2006 Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010

Microwave Network Analysis

A. Nassiri -ANL

Lecture 7

The US Particle Accelerator School Boston University June 16th, 2006 Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 2

S-Parameter Measurement Technique

VVM: The vector voltmeter measures the magnitude of areference and test voltage and the difference in phase between thevoltages. Because it can measure phase, it allows us to directlymeasure the S-parameters of a circuit

Unfortunately, the use of the directional couplers and test cablesconnecting the measuring system to the vector voltmeterintroduces unknown attenuation and phase shift into themeasurements. These can be compensated for by makingadditional “calibration” measurements.


Signal Gen VVMA B

N-BNC N-BNC N-BNC

N-BNC

N-BNC N-BNC

BNC cable

BNC cableDUT

20 dB 20 dB

Matched load

(HP 8657B) (HP 8508A)

(HP 778D Dual Directional Coupler)

Reflection measurements: S11 or S22


From the setup, it is seen that the voltage at channel A of the VVM (AD)is proportional to the amplitude of the voltage wave entering the deviceunder test (DUT) (aD

1). Similarly, the voltage at channel B (BD) isproportional to the amplitude of the voltage wave reflected from DUT(bD

1). Thus we can write

DB

D

DA

D

bKB

aKA

1

1

=

=

Where KA and KB are constants that depend on the connecting cables.Since aD

2 is zero because of the matched load at port 2, S11 is given by

AD

BD

D

D

KAKB

ab

S ==1

111



To find KA and KB it is necessary to make a second measurement with aknown DUT. This is called a “calibration” measurement. If the DUT isremoved and replaced by a short circuit, the voltage at channel A (As)and channel B(Bs) are given by

SB

S

SA

S

bKB

aKA

1

1

=

=

Where as1 is the amplitude of the voltage wave entering the short and bs

1

is the amplitude of the voltage wave reflected from the short. However,for a short circuit the ratio of these amplitudes is –1 (reflectioncoefficient of a short). Thus

11

1 −==A

SB

S

S

S

KAKB

ab



S

S

A

B

AB

KK

−=

−=

S

S

D

D

AB

AB

S11

Note: since VVM displays quantities in terms of magnitude and phase we can rewrite S11 as

( )π−φ−φ∠ΓΓ

= SDS

DS11

DDD

D

AB

φ∠Γ=

SSS

S

AB

φ∠Γ=



Transmission measurements: S12 or S21

N-BNC

BNC cable

DUT

Generator Channel A Channel BMatched load Matched load

Matched load

The DUT is connected directly between two directional couplers.Voltage at A of the VVM is proportional to the voltage wave incident onthe DUT while the voltage at B of the VVM is proportional to voltagewave transmitted through the DUT.


DB

D

DA

D

bKB

aKA

2

1

=

=

AD

BD

D

D

KAKB

ab

S ==1

221⇒

To find out the constants a calibration measurement must be made.Remove the DUT and connect both directional couplers directlytogether. The Known DUT in this case is just a zero-length guide with atransmission coefficient of unity. The measured voltages are:

EB

E

EA

E

bKB

aKA

2

1

=

= 11

2 ==A

EB

E

E

E

KAKB

ab

where

E

E

A

B

AB

KK

=∴




−=

E

E

D

D

AB

AB

S 21 ( )ETT

S DE

Dθ−θ∠=21

DDD

DT

AB

θ∠=

EEE

ET

AB

θ∠=

where



Scattering Parameters

Scattering Parameters (S-Parameters) plays a major role is networkanalysis

This importance is derived from the fact that practical systemcharacterizations can no longer be accomplished through simple open- orshort-circuit measurements, as is customarily in low-frequencyapplications.

In the case of a short circuit with a wire; the wire itself possesses aninductance that can be of substantial magnitude at high frequency.

Also open circuit leads to capacitive loading at the terminal.


In either case, the open/short-circuit conditions needed to determineZ-, Y-, h-, and ABCD-parameters can no longer be guaranteed.

Moreover, when dealing with wave propagation phenomena, it is notdesirable to introduce a reflection coefficient whose magnitude is unity.

For instance, the terminal discontinuity will cause undesirable voltageand/or current wave reflections, leading to oscillation that can result inthe destruction of the device.

With S-parameters, one has proper tool to characterize the two-portnetwork description of practically all RF devices without harm to DUT.



Definition of Scattering Parameters

S-parameters are power wave descriptors that permit us to define theinput-output relations of a network in terms of incident and reflectedpower waves.

a1 a2

b1 b2

[S]

an – normalized incident power waves

bn – normalized reflected power waves


( ) ( )12

1 nnn IZVZ

a

+=

( ) ( )22

1 nnn IZVZ

b

−=

Index n refers either to port number 1 or 2. The impedance Z0 is thecharacteristic impedance of the connecting lines on the input and outputside of the network.



Inverting (1) leads to the following voltage and current expressions:

( ) ( )3 nnn baZV +=

( ) ( )41 nnn baZ

I −=



Recall the equations for power:

( )5 21

21 22

−== nn

*nnn baIVReP

Isolating forward and backward traveling wave components in (3) and(4), we see

( )

( )7

6

−−

++

−==

==

nn

n

nn

n

IZZ

Vb

IZZ

Va



We can now define S-parameters:

( )82

1

2221

1211

2

1

=

aa

SSSS

bb



( )

( )

( )

( )12 2port at power waveIncident

1port at wavepowe dTransmitte

11 2port at power waveIncident 2port at wavepowe Refkected

10 1port at power waveIncident

2port at wavepowe dTransmitte

9 2port at power waveIncident 1port at wavepowe Refkected

02

112

02

222

01

221

01

111

1

1

2

2

==

==

==

==

=

=

=

=

a

a

a

a

abS

abS

abS

abS



Observations

a2=0, and a1=0 ⇒ no power waves are returned to the network ateither port 2 or port 1.

However, these conditions can only be ensured when the connectingtransmission line are terminated into their characteristic impedances.

Since the S-parameters are closely related to power relations, we canexpress the normalized input and output waves in terms of timeaveraged power.

The average power at port 1 is given by

( ) ( ) ( )131211

21 2

11

212

21

1 SZ

V

Z

VP in −=Γ−=

++



The reflection coefficient at the input side is expressed in terms of S11under matched output according:

( ) 141101

1

1

1

2

Sab

VV

ain ===Γ

=+

−

This also allow us to redefine the VSWR at port 1 in terms of S11 as

( )1511

11

11 SS

VSWR −+

=


We can identify the incident power in (13) and express it in terms ofa1:

( )1622

1 21

21

a

PZ

Vinc ==

+

Maximal available powerfrom the generator

The total power at port 1 (under matched output condition) expressedas a combination of incident and reflected powers:

( ) ( ) ( )17122

1 22

121

211 inreflinc

abaPPP Γ−=−=+=



If the reflected coefficient, or S11, is zero, all available power from thesource is delivered to port 1 of the network. An identical analysis at port2 gives

( ) ( ) ( )18122

1 22

222

222 out

abaP Γ−=−=



Meaning of S-Parameters

S-parameters can only be determined under conditions of perfectmatching on the input or the output side.

ZL

a1

b1 b2

[S]

a2=0

VG1 Zo Zo

Zo

Measurement of S11 and S21 by matching the line impedance Zo

at port 2 through a corresponding load impedance ZL=Zo


This configuration allows us to compute S11 by finding the inputreflection coefficient:

( )1911

ZZZZ

Sin

inin +

−=Γ=

Taking the logarithm of the magnitude of S11 gives us the return lossin dB

( )2020 11 SRL log−=



With port 2 properly terminated, we find

( ) ( ) ( )212

011

2

01

221

222

==

−

= +++==

VIa ZIZVZV

ab

S

Since a2=0, we can set to zero the positive traveling voltage andcurrent waves at port 2.

Replacing V1 by the generator voltage VG1 minus the voltage dropover the source impedance Zo, VG1-ZoI1 gives

( )22221

2

1

221

GG VV

VV

S ==−



The forward power gain is

( )232

2

1

2221

GVV

SG ==

If we reverse the measurement procedure and attach a generatorvoltage VG2to port 2 and properly terminate port 1, we can determine theremaining two S-parameters, S22 and S12.

a1=0

b1 b2

[S]

a2

VG2Zo ZoZo

Zo



To compute S22 we need to find the output reflection coefficient Γout ina similar way for S11:

( )2422

ZZZZ

Sout

outout +

−=Γ=

( ) ( ) ( )252

022

1

02

112

111

==

−

= +++==

VIa ZIZVZV

ab

S

( ) ( ) 272

26222

2

1212

2

1

2

112

Gor

GG VV

SGV

VVV

S ====−

Reverse power gain



Determination of a T-network elements

Find the S-parameters and resistive elements for the 3-dBattenuator network. Assume that the network is placed into atransmission line section with a characteristic lineimpedance of Zo=50 Ω

R1 R2

R3Port 1 Port 2


An attenuator should be matched to the line impedance andmust meet the requirement S11= S22= 0.

R1 R2

R3

Port 1 Port 2

50Ω

Circuit for S11 and S21

( )( )

RRRR

RZ in Ω=Ω++

Ω++= 50

5050

23

231

Because of symmetry, it is clear that R1=R2.



We now investigate the voltage V2 = V-2 at port 2 in

terms of V1=V+1.

R1 R2

R3

Port 1 Port 2

50Ω

( )( )

( )( )

11

113

13

13

13

2 5050

5050

5050

VRR

RRRR

RRRR

V

+ΩΩ

+Ω++

Ω+Ω++

Ω+

=



For a 3 dB attenuation, we require

121

2

1

221 7070

212

SVV

VV

SG

===== .

Setting the ratio of V2/V1 to 0.707 and using theinput impedance expression, we can determine R1

and R3

Ω==

Ω=+−

==

414122

5881212

3

21

.

.

ZR

ZRR



Note: the choice of the resistor network ensures that at theinput and output ports an impedance of 50 Ω is maintained.This implies that this network can be inserted into a 50 Ωtransmission line section without causing undesiredreflections, resulting in an insertion loss.



Chain Scattering Matrix

To extend the concept of the S-parameter presentation tocascaded network, it is more efficient to rewrite the power waveexpressions arranged in terms of input and output ports. This resultsin the chain scattering matrix notation. That is,

( )282

2

2221

1211

1

1

=

ab

TTTT

ba

It is immediately seen that cascading of two dual-port networksbecomes a simple multiplication.


Aa1

Ab1

Aa2

Ab2

Bb1

Bb2

Ba2

Ba1

[ ]AT [ ]BTPort 1 Port 2

Cascading of two networks A and B



If network A is described by

( )292

2

2221

1211

1

1

=

A

A

AA

AA

A

A

ab

TTTT

ba

And network B by

( )302

2

2221

1211

1

1

=

B

B

BB

BB

B

B

ab

TTTT

ba



( )311

1

2

2

=

B

B

A

A

ba

ab

Thus, for the combined system, we conclude

( )312

2

2221

1211

2221

1211

1

1

=

B

B

BB

BB

AA

AA

A

A

a

b

TT

TT

TT

TT

b

a



The conversion from S-matrix to the chain matrix notation is similaras described before.

( )32121121

1

02

111

2

SaS

aba

Ta

====

( )3321

2212

SS

T −=

( )3421

1121

SS

T =

( ) ( )352121

2112221122 -

SS

SSSSS

T∆

=−−

=



Conversely, when the chain scattering parameters are given and weneed to convert to S-parameters, we find the following relations:

( )

( ) ( )

( )

( )39

381

37

36

11

1222

1121

1111

2112221112

11

21

211

221

02

111

2

TT

S

TS

TT

TTTTT

S

TT

bTbT

ab

Sa

−=

=

∆=

−=

====



Conversion between Z- and S-Parameters

To find the conversion between the S-parameters and the Z-parameters,let us begin with defining S-parameters relation in matrix notation

[ ] ( )40 aSb =Multiplying by gives

Z

[ ] [ ] ( )41 +− === VSaSZVbZ

Adding to both sides results in aZV =+

[ ] [ ] [ ]( ) ( )42 +++ +=+= VESVVSV


To compare this form with the impedance expression

[ ] IZV = We have to express V+ in term of I. Subtract [SV+ from both sides of

aZV =+

[ ] ( ) ( )43 IZbaZVSV =−=− ++

[ ] [ ]( ) ( )44 1 ISEZV −+ −=



Substituting (44) into (42) yields

[ ] [ ]( ) [ ] [ ]( ) [ ] [ ]( ) ( )45 1 ISEESZVESV −+ −+=+=

or

[ ] [ ] [ ]( ) [ ] [ ]( ) ( )461 −−+= SEESZZ

Explicitly

( )( )( )47

11

111

11

11

1

1121

1222

12212211

2221

1211

1

2221

1211

2221

1211

2221

1211

−

−−−−

+

+

=

−−−−

+

+=

−

SSSS

SSSSSS

SSZ

SSSS

SSSS

ZZZZZ



Practical Network Analysis


Linear Versus Nonlinear Behavior

Linear behavior:input and output frequencies are the

same (no additional frequencies created)output frequency only undergoes

magnitude and phase change

Frequencyf1

Time

Sin 360 * f * t°

Frequency

Aphase shift = to * 360 * f°

1f

DUT

Time

A

to

A * Sin 360 * f ( t - t )° °

Input Output

Time

Nonlinear behavior:output frequency may undergo frequency

shift (e.g. with mixers)additional frequencies created

(harmonics, inter-modulation)

Frequencyf1


Criteria for Distortionless TransmissionLinear Networks

Constant amplitude over bandwidth of interest

Mag

nitu

de

Frequency

Pha

se

Frequency

Linear phase over bandwidth of interest


Magnitude Variation with Frequency

Linear Network

Frequency Frequency

Mag

nitu

de

Frequency

Time Time

ttttf ω+ω+ω= 5513

31 sinsinsin)(


Phase Variation with Frequency

Frequency

Mag

nitu

de

Frequency

Frequency

Time

Linear Network

0

-180

-360

°

°

°

Time

ttttf ω+ω+ω= 5513

31 sinsinsin)(


Criteria for Distortionless TransmissionNonlinear Networks

Frequency

Time Time

Saturation, crossover, inter-modulation, and other nonlinear effects can cause signal distortion

Frequency


The Need for Both Magnitude and Phase

Complex impedance needed to design matching circuits

Complex values needed for device modeling

4. Time Domain Characterization

Mag

Time

Complete characterization of linear networks

1.

2.

3.

S

S

S 11

21

22

12

S

Vector Accuracy Enhancement5.

High Frequency Transistor Model

CollectorBase

Emitter

Error

Measured

Actual


High-Frequency Device Characterization

Incident

Reflected

Transmitted

Lightwave Analogy


Transmission Line Review

Low frequenciesWavelength >> wire lengthCurrent (I) travels down wires easily for efficient power transmissionVoltage and current not dependent on position

High frequenciesWavelength ≈ or << wire (transmission line) lengthNeed transmission-line structures for efficient power transmissionMatching to characteristic impedance (Z0)

is very important for low reflectionVoltage dependent on position along line

I


Transmission Line Terminated with Zo

For reflection, a transmission line terminated in Zobehaves like an infinitely long transmission line

Zs = Zo

Zo

Vrefl = 0! (all the incident power is absorbed in the load)

Vinc

Zo = characteristic impedance of transmission line


Transmission Line Terminated with Short, Open

Zs = Zo

Vrefl

Vinc

For reflection, a transmission line terminated in a short or open reflects all power back to source

In phase (0 ) for openOut of phase (180 ) for short

oo


Transmission Line Terminated with 25Ω

Zs = Zo

ZL = 25 Ω

Vrefl

Vinc

Standing wave pattern does not go to zero as with short or open


High-Frequency Device Characterization

SWR

S-ParametersS11,S22 Reflection

Coefficient

Impedance, Admittance

R+jX, G+jB

ReturnLoss

Γ, ρ

Incident

Reflected

TransmittedR BA

ReflectedIncident

REFLECTION

AR=

TransmittedIncident

TRANSMISSION

BR=

Gain / Loss

S-ParametersS21,S12

GroupDelay

TransmissionCoefficient

Insertion Phase

Τ,τ


Reflection Parameters

=ZL − ZO

ZL + OZReflection

Coefficient=

Vreflected

Vincident= ρ ΦΓ

ρ

ρRL

VSWR

No reflection(ZL = Zo)

Full reflection(ZL = open, short)

∞ dB0

1

1

0 dB

∞

= ΓReturn loss = -20 log(ρ),

VSWR = Emax

Emin=

1 + ρ1 - ρ

Voltage Standing Wave RatioEmaxEmin


Transmission Parameters

Transmission Coefficient = Τ =VTransmitted

VIncident= τ∠φ

V TransmittedV IncidentDUT

Insertion Loss (dB) = - 20 Log V Trans

V Inc = - 20 log τ

Gain (dB) = 20 Log V Trans

V Inc

= 20 log τ


Deviation from Linear Phase

Use electrical delay to remove linear portion of phase response

+ yields

Linear electrical length added

Frequency

(Electrical delay function)RF filter response

Phas

e 45

/D

ivo

Frequency Frequency

Deviation from linear phase

Phas

e 1

/Div

o

Low resolution High resolution


Low-Frequency Network Characterization

H-parametersV1 = h11I1 + h12V2

V2 = h21I1 + h22V2

Y-parametersI1 = y11V1 + y12V2

I2 = y21V1 + y22V2

Z-parametersV1 = z11I1 + z12I2

V2 = z21I1 + z22I2

h11 = V1

I1 V2=0

h12 = V1

V2 I1=0

(requires short circuit)

All of these parameters require measuring voltage and current (as a function of frequency)

(requires open circuit)


Limitations of H, Y, Z Parameters(Why use S-parameters?)

H,Y, Z parametersHard to measure total voltage and current at

device ports at high frequenciesActive devices may oscillate or self-destruct with shorts opens

S-parametersRelate to familiar measurements

(gain, loss, reflection coefficient ...)Relatively easy to measureCan cascade S-parameters of multiple

devices to predict system performanceAnalytically convenient

CAD programsFlow-graph analysis

Can compute H, Y,or Z parameters from S-parameters if desired

Incident TransmittedS21

S11Reflected S22

Reflected

Transmitted Incidentb1

a1 b2

a 2S12

DUT

b1 = S11 a1 +S12 a 2b2 = S21 a1 + S22 a 2

Port 1 Port 2


Measuring S-Parameters

S 11 = ReflectedIncident

=b1a 1 a2 = 0

S 21 =Transmitted

Incident=

b2

a 1 a2 = 0

S 22 = ReflectedIncident

=b2a 2 a1 = 0

S 12 =Transmitted

Incident=

b1

a 2 a1 = 0

Incident TransmittedS 21

S 11Reflected

b1

a 1

b2

Z 0Load

a2 = 0DUTForward

1IncidentTransmitted S 12

S 22Reflected

b2

a2

b

a1 = 0DUTZ 0

LoadReverse


What is the difference between

network and spectrum analyzers?.

Network analyzers: measure components, devices, circuits,

sub-assemblies contain source and receiver display ratioed amplitude and phase

(frequency or power sweeps)

Spectrum analyzers: measure signal amplitude characteristics

(carrier level, sidebands, harmonics...) are receivers only (single channel) can be used for scalar component test (no

phase) with tracking gen. or ext. source(s)

Ampl

itude

Rat

io

Frequency

Hard: getting (accurate) traceEasy: interpreting results

Measures known signal

Pow

er

Frequency

Easy: getting traceHard: interpreting results

8563A SPECTRUM ANALYZER 9 kHz - 26.5 GHz

Measures unknown

signals


Signal Separation

Measuring incident signals for ratioing

Splitterusually resistivenon-directionalbroadband

6 dB50 Ω

50 Ω 6 dB

Couplerdirectionallow lossgood isolation, directivityhard to get low freq performance

Main signal

Coupled signal


Forward Coupling Factor

Coupling Factor (dB) = -10 logPcoupling forward

Pincident

Source

Z0

0 dBm1 mW

-.046 dBm.99 mW

-20 dBm.01 mW

Coupling, forward

Example of 20 dB Coupler


Directional Coupler Isolation(Reverse Coupling Factor)

Source

Z 0

0 dBm 1 mW

−.046 dBm.99 mW

-50 dBm.00001 mW

Coupling, reverse

Example of 20 dB Coupler "turned around"

Isolation Factor (dB) = -10 logPcoupled reverse

Pincident

this is an error signal during measurements


Directional Coupler Directivity

Directivity (dB) = 10 log Pcoupled forward

Pcoupled reverse

Directivity =Coupling Factor

Isolation

Directivity (dB) = Isolation (dB) - Coupling Factor (dB)

Example of 20 dB Coupler with 50 dB isolation:Directivity = 50 dB - 20 dB = 30 dB


Measuring Coupler Directivity the Easy Way

short

1.0 (0 dB) (reference)

Coupler Directivity

35 dB

load Assume perfect load

.018 (35 dB) (normalized)Source

Source

Good approximation for coupling factors ≥ 10 dB

Directivity = 35 dB - 0 dB = 35 dB


Narrowband Detection - Tuned Receiver

Best sensitivity / dynamic rangeProvides harmonic / spurious signal rejectionImprove dynamic range by increasing power, decreasing IF bandwidth, or averagingTrade off noise floor and measurement speed

ADC / DSP

10 MHz 26.5 GHz


Comparison of Receiver Techniques

0 dB

-50 dB

-100 dB

-60 dBm Sensitivity

Broadband (diode) detection

< -100 dBm Sensitivity

0 dB

-50 dB

-100 dB

Narrowband (tuned-receiver) detection

higher noise floorfalse responses

high dynamic rangeharmonic immunity

Dynamic range = maximum receiver power - receiver noise floor

The US Particle Accelerator School Boston University June 16th, 2006

Dynamic Range and AccuracyDynamic range is very important for measurement accuracy !

Error Due to Interfering Signal

0 -5 -10 -15 -20 -25 -30 -35 -40 -45 -50 -55 -60 -65 -700.001

0.01

0.1

1

10

100

Interfering signal (dB)

Error (dB, deg)

+ magn (dB)

- magn (dB)

phase (± deg)phase error

magn error


Measurement Error Modeling

Systematic errorsdue to imperfections in the analyzer and test setupare assumed to be time invariant (predictable)can be characterized (during calibration process) and mathematically removed during measurements

Random errorsvary with time in random fashion (unpredictable)cannot be removed by calibrationmain contributors:

instrument noise (source phase noise, IF noise floor, etc.)

switch repeatabilityconnector repeatability

Drift errorsare due to instrument or test-system performance

changing after a calibration has been doneare primarily caused by temperature variationcan be removed by further calibration(s)

Unknown Device

Measured Data

SYSTEMATIC

RANDOM

DRIFT

Errors:


Systematic Measurement Errors

A B

SourceMismatch

LoadMismatch

CrosstalkDirectivity

DUT

Frequency response reflection tracking (A/R) transmission tracking (B/R)

R

Six forward and six reverse error terms yields 12 error terms for two-port devices


Types of Error CorrectionTwo main types of error correction:

response (normalization)• simple to perform• only corrects for tracking errors• stores reference trace in memory,then does data divided by memory

vector• requires more standards• requires an analyzer that can measure phase• accounts for all major sources of systematic error

S11M

S11A

SHORT

OPEN

LOAD

thru

thru


ABCD parameter representationVery useful when cascading networks

i`2 i``

1

v`2 v``

1

i``2

v``2

+

- -

+

′′′′

DCBA

′′′′′′′′

DCBA

Port 1 Port 2

′′−ν ′′

′′′′′′′′

′′′′

=

ν

2

2

1

1

iDCBA

DCBA

i

-

i`1

i`2 i``

1

i``2

v`1 v`

2 v``1

v``2

++

+

- -

-

i`1

i`2 i``

1

i``2

v`1 v`

2 v``1

v``2

++

+

- -


ABCD parameter representationABCD very useful for T.L.

li1 i2

v1 v2Z0, β( ) ( )( ) ( )

ββ

ββ=

ll

ll

cossinsincos

Zj

jZ

DCBA

Z0, βl

R

L L

C

Z0, βl

R

L L

C

Input matching Network

Feedback loopOutput matching Network


Signal Flow ComputationsComplicated networks can be efficiently analyzed in a manner identical to signals and systems and control.

Z0

a

b

ZL

a

b

ΓL

Γijai bj

in general


Signal Flow Graphs

Basic Rules:We’ll follow certain rules when we build up a network flow graph.

1. Each variable, a1, a2, b1, and b2 will be designated as a node.

2. Each of the S-parameters will be a branch.

3. Branches enter dependent variable nodes, and emanate from the independent variable nodes.

4. In our S-parameter equations, the reflected waves b1 and b2 are the dependent variables and the incident waves a1 and a2 are the independent variables.

5. Each node is equal to the sum of the branches entering it.


Let’s apply these rules to the two S-parameters equations

2221212

2121111

aSaSb

aSaSb

+=+=

First equation has three nodes: b1, a1, and a2. b1 is a dependent node and is connected to a1 through the branch S11 and to node a2 through the branch S12. The second equation is similar.

a1

a2b1

S11

S22

a1 b2

a2

S22

S21

Signal Flow Graphs


Complete Flow Graph for 2-Port

a1

a2b1

S11

S12

S21

S11

b2

The relationship between the traveling waves is now easily seen. Wehave a1 incident on the network. Part of it transmits through the networkto become part of b2. Part of it is reflected to become part of b1.Meanwhile, the a2 wave entering port two is transmitted through thenetwork to become part of b1 as well as being reflected from port two aspart of b2. By merely following the arrows, we can tell what’s going on inthe network. This technique will be all the more useful as we cascadenetworks or add feedback paths.

Signal Flow Graphs


Arrangement for Signal Flow Analysis

VG

ZG

Z0 ZL

b’

a’

a

b

IG

ba’

bs b’ a

ΓS ΓL

GG

s VZZ

Zb

+=

bs a

b

ΓS ΓL


Analysis of Most Common Circuit

VS

ZS

Z0 ZL

b1

Z0[S]

b2

bs a1 a2ΓS ΓL

bs

ΓS ΓL

a1

a2b1

b2

S11 S22

S21

S12

bs a1

SLL

s

SSS

Sba

Γ

Γ

Γ−+−

→

22

211211

1

11

1


S

SSS

abΓ L

Lin Γ

Γ−+==

22

211211

1

11

a1

bs

ΓS

b1

S11

LLS

SSΓ

Γ− 22

21121

bs

ΓS

b1

a1

LLS

SSS Γ

Γ−+

22

211211 1

Note: Only = 0 ensures that S11 can be measured. LΓ


Scattering Matrix

The scattered-wave amplitudes are linearly related to the incident wave amplitudes. Consider the N port junction

V+1

V-1

V+2

V-2

V+3

V-3

V+4

V-4

V+N

V-N

Port 1Port 2

Port 3

Port 4

Port N

Port 5

V+5

V-5

If the only incident wave is V+1 then

+− = 1111 VSV

11S is the reflection coefficient

The total voltage is port 1 is −+ += 111 VVV

Waves will also be scattered out of other ports. We will have

NnVSV nnn ,...,, 4321 == +−


If all ports have incident wave then

=

+

+

+

−

−

−

NNNNNN

N

N

N V

VV

SSSS

SSSSSSSS

V

VV

......

.....................

...2

1

321

2232221

1131211

2

1

or

[ ] [ ][ ]+− = VSV

[ ]S is called the scattering matrix ( )jkVforvv

S kj

iij ≠== +

+

− 0

Scattering Matrix


If we choose the equivalent Z0 equal to 1 then the incident power is given by

2

21 +

nV

and the scattering will be symmetrical. With this choice

−+−+ +=+= IIIVVV ,

( )

( )IVV

IVV

and

−=

+=

−

+

2121

Scattering Matrix


V+ and V- are the variables in the scattering matrix formulation; but they are linear combination of V and I.

Other normalization are

ZV

=νZ

Ii =

Just as in the impedance matrix there are several properties of the scattering matrix we want to consider.

1. A shift of the reference planes

2. S matrix for reciprocal devices

3. S matrix for the lossless devices

Scattering Matrix


Shift in the reference planesConsider the following network, where tn is the original location of the reference plane, and t’

n in the new location of the reference plane.

ntnt ′

ln

The electrical length between and is

.

nt nt ′

nn lβ=θ

n

n

jnn

jmn

eS

enmSθ−

θ−≠2 bylied be multip must

bylied be multip must

Why is this a factor of 2?

Scattering Matrix


n

n

jnn

jnn

eVV

eVVθ−−−

θ++

=′

=′

[ ]

=

θ−

θ−

θ−

θ−

θ−

θ−

NN j

j

j

nnNN

N

N

j

j

j

e

e

e

SSS

SSSSSS

e

e

e

S

. .

. .

2

1

2

1

21

22221

11211

...............

...

...

Scattering Matrix


Consider a 2×2 scattering matrix where two reference planes are shifted.

[ ]

[ ]

[ ]

=′

=′

=′

θ−θ−θ−

θ−θ−θ−

θ−θ−θ−

θ−θ−θ−

θ−

θ−

θ−

θ−

221

211

221

211

2

1

2

1

22212

212

11

2221

1211

2221

1211

00

00

jjj

jjj

jjj

jjj

j

j

j

j

eSeeSeeSeS

S

eSeeSeeSeS

S

ee

SSSS

ee

S

Scattering Matrix


Proof of Symmetry of Scattering MatrixFor a reciprocal junction Smn = Snm m≠n provided that Z0=1

2

21 += nVP for all modes.

( )1=−=∴

−=+=−+

−+−+

ZVVI

IIIandVVV

nnn

nnnnnn

[ ] [ ] [ ]−+ += VVV

[ ][ ][ ][ ] [ ][ ]−+ −=

=

VZVZ

IZ

Scattering Matrix


Defining the unit matrix

[ ]

=

10

101

.u

[ ] [ ] [ ][ ] [ ][ ]−+−+ −=+ VZVZVV

[ ] [ ]( )[ ] [ ] [ ]( )[ ]+− −=+ VUZVUZ

Rearrange and factor

or

[ ] [ ] [ ]( ) [ ] [ ]( )[ ]+−− −+= VUZUZV 1

[ ] [ ][ ]+− = VSVbut [ ] [ ] [ ]( ) [ ] [ ]( )UZUZS −+=∴ −1

Scattering Matrix


Loss-less JunctionThe total power leaving a junction must be equal to the total power entering the junction power.

2

1

2

1∑=∑=

+

=

− N

nn

N

nn VV

so that

+

=

− ∑= i

N

inin VSV

1

but

2

11

2

1∑=∑ ∑=

+

=

+

=

N

nn

N

ni

N

ini VVS

Scattering Matrix


Loss-less JunctionV+

n are all independent incident voltages, so we choose V+n =0 except for

n=i

.* iSSS

VVS

VVS

ni

N

nni

N

nni

n

N

nini

N

nn

N

ni

N

ini

∀=∑=∑

=∑

∑=∑ ∑

==

+

=

+

=

+

=

+

=

111

2

22

1

2

11

2

1

[ ] 12

1

21 =

*

*

*

......

Ni

i

i

Niii

S

SS

SSS

Scattering Matrix


Loss-less 2 Port Junction

For this case [S] is unitary

mpnp

N

nnmSS δ=∑∴

=

*

1

or

0

1

1

22121211

221112122222

12121111

=+

=⇒

=+

=+

**

**

**

SSSS

SSSSSS

SSSS

The magnitude of the input and output ports are equal in magnitude. Also

21112 1 SS −=

Scattering Matrix


If we know then we can obtain and . 11S 12S 22S

Note: The fraction of power reflected at terminal t, is

211S

PP

inc

refl =

So that the insertion loss due to reflection is

( )12

211

20110SIL

SIL

loglog

=

−=

Scattering Matrix


Example: two-port network

l1

l2

t1

t2

Assume TE10 modes at t1 and t2

Equivalent Circuit

V +2

+1V

Z1 Z2

Z31I 2I

Port 1 Port 2

Apply KVL:

1323222

2313111

IZIZIZV

IZIZIZV

++=++=

Scattering Matrix


If

12222

12111

02

1123

1

ZZZ

ZZZ

IV

ZZI

−=−=

===

Then we have

2122222

2121111

IZIZV

IZIZV

+=+=

and

[ ] [ ][ ]IZV =

1211 ZZ − 1222 ZZ −

12Z

Scattering Matrix


This can be transformed into an admittance matrix

=

2

1

2212

1211

2

1

VV

YYYY

II

1211 YY − 1222 YY −

12Y

Scattering Matrix


Traveling Wave:

( ) ( ) ( )xVxVxV

AeVAeV xx

−+

δ−δ−+

+=

== ,

Similarly for current:

( ) ( ) ( ) ( ) ( ) ZxV

ZxV

xIxIxI−+

−+ −=−=

Reflection Coefficient:

( ) ( )( )xVxV

x +

−=Γ

Scattering Matrix


Introduce “normalized” variables:

( ) ( ) ( ) ( )xIZxZxVx ==ν i,

So that

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )xaxxbxbxaxxbxax Γ=−=+=ν i and

This defines a single port network. What about 2-port?

2-port

2221212

2121111

aSaSb

aSaSb

+=+=

Scattering Matrix


Each reflected wave (b1,b2) has two contributions: one from the incident wave at the same port and another from the incident wave at the other port.

How to calculate S-parameters?

01

111

2 =

=aa

bS

012

112

=

=aa

bS

01

221

2 =

=aa

bS

02

222

1=

=aa

bS

Input reflected coefficient with output matched.

Reverse transmission coefficient with input matched.

Transmission coefficient with output matched.

Output reflected coefficient with input matched.

Scattering Matrix

The US Particle Accelerator School Boston University June 16th, 2006 Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010

Generalized Scattering Matrix:We define it via [b]=[S][a]. S-matrix depends on the choice of normalized impedance. Usually 50 Ω, but can be anything and can even be complex!

Calculating Sij:

ii

ii

iii

iii

nkikai

iij ZZ

ZZ

IZV

IZVab

Sk ,

*,

*,

*,

,...,,

+−

=+

−==

=≠= 10

Which is input reflected coefficient with all other ports matched.

nkikaiki

kabk

S,...,, 10 =≠=

=

is equal to transducer power gain from i to k with ports other than i matched.

Scattering Matrix

microwave network analysis - uspas.fnal.gov › materials › 10mit › lecture7.pdf · microwave...

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